Problem Description

The problem states that an integer x is considered good if it can be turned into a valid number by rotating each digit by 180 degrees. The rotation should transform the number into a different one, meaning we can't just leave any digit unchanged.

A digit is valid after rotation if it's still a number. There are specific digits that transform into themselves or others:

• 0, 1, and 8 rotate to themselves.
• 2 and 5 can rotate to each other.
• 6 and 9 also rotate to each other.

Digits that are not in the above list become invalid after rotation.

Our task is to find the total number of good integers within the range from 1 up to and including n.

Intuition

To solve this problem, we approach it through recursive depth-first search (DFS) with caching (memoization) to improve performance. The idea is to iterate through each digit position, determining whether placing a certain digit there would contribute to the count of good numbers.

We define a recursive function dfs that considers the following arguments:

• pos: current position in the number we're inspecting.
• ok: a boolean indicating whether we've already placed a digit that becomes different after rotation (making the current number formation a good number).
• limit: a boolean that tells us if we are limited by the original number n in choosing our digits.

The base case of the recursion is when pos <= 0, which means we've checked all the digit positions. If ok is True at this point, we've formed a good number, and we can increment our count.

During each recursive call, we try placing each digit (0-9) in the current position and make further recursive calls to fill the remaining positions. There are conditions though:

• If the digit is 0, 1, or 8, we can continue the recursion without changing the ok status, as these digits do not alter the number's goodness by themselves.
• If the digit is 2, 5, 6, or 9, we make the recursive call with ok set to True, since these digits ensure the number's goodness.

To avoid unnecessary calculations, we memoize the results of the recursive calls with different parameter combinations.

We structure the number n into an array a, where each element represents a digit in its corresponding place. Then we walk through the number's digits from the most significant digit to the least significant digit (right to left), calling our recursive function to sum up the counts of good numbers according to the above-defined rules.

The solution's effectiveness comes from carefully analyzing the properties of good numbers and utilizing recursive calls with caching to systematically count the valid numbers without enumerating them all.

Learn more about Math and Dynamic Programming patterns.

Solution Approach

The provided Python solution makes use of recursion, dynamic programming through memoization, and digit-by-digit analysis to solve the problem of finding good integers within a given range.

Firstly, a recursive helper function dfs is defined. This function uses the following parameters:

• pos: The current digit position we are filling in the potentially good integer.
• ok: A boolean flag indicating whether we've already included at least one digit in the number that makes it 'good' upon rotation, such as 2, 5, 6, or 9.
• limit: A flag that indicates whether the current digit being considered is constrained by the original integer n's corresponding digit.

The dfs function utilizes memoization to cache and reuse the results of the recursive calls, which prevents recalculating the same scenarios—this is done using the @cache decorator, which caches the results of the expensive function calls.

The recursive calls made by the dfs function follow these rules:

• If pos is 0 or negative, it means we've looked at all digit positions, and we check the flag ok. If ok is true, we return 1 as we've found a good number; otherwise, we return 0 as the number does not meet the criterion.
• Otherwise, we iterate over all possible digits from 0 to 9 (or up to the digit in the original number n if the limit is true), placing each digit in the current position and making a recursive call to dfs to decide the next position.
• The current state of ok is passed on unless we're placing a digit that contributes to the goodness of the number (2, 5, 6, or 9), in which case ok is set to true for all deeper recursive calls.
• The limit is only maintained if the digit being placed is equivalent to the corresponding digit in n.

To kick off the recursion, the original number n is broken down into its constituent digits and stored in an array a. This allows the function to easily compare each possible combination against n's corresponding digits to ensure validity.

Finally, the dfs function is called with the array length, ok set to 0 (as initially, we haven't placed any digits that would classify the integer as good), and limit set to True (because at the start, we are limited to n). The return value from this call gives the total number of good integers within the range specified.

This solution effectively parses the problem domain with a depth-first traversal, leveraging the constraints to prune the search space and using memoization to optimize the recursive exploration of possibilities, resulting in an efficient algorithm to find all good integers up to n.

Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we want to find all good integers up to n = 23. Using the definitions and rules from the solution approach, we would proceed as follows:

1. We start by setting up our dfs function and decomposing our target integer n = 23 into an array a = [2, 3].

2. We then call dfs(pos=len(a), ok=False, limit=True) to start the recursion, which implies we are starting from the digit in the tens place (as array indices are 0-based).

3. At the first level of recursion (pos = 2), we have not yet placed any digit, so ok is False, and limit is True. We iterate over possible digits from 0 to 2 for the tens place since 2 is the tens digit of n (due to limit=True).

   • When we place a 0 or 1, we keep ok as False since these digits don't contribute to a number’s goodness. We also set limit to False for the next digit since we are already below n.
   • When we place a 2 (same as in n), we keep limit as True for the next digit to make sure we don't go over n. Here, we set ok to True because 2 contributes to the number’s goodness upon rotation (2 rotates to 5).

4. Now, for the second level of recursion (pos = 1), we are checking the digit in the ones place. The limit flag instructs us whether we can consider digits up to 3 or all digits 0-9.

   • If we have placed a 0 or 1 in the tens place, we can consider all digits here, and if we place a 2, 5, 6, or 9, we set ok to True.
   • If we had placed a 2 in the tens place, we can go up to 3 in the ones place.

5. Each time we reach pos = 0 (the base case), we check if ok is True, meaning we have made the number good. If so, we return 1; otherwise, we return 0.

6. Finally, after considering all possibilities for each digit, the sum of all recursive call returns will give us the count of good integers. In our case, we would find the valid good numbers to be 2, 5, 6, 8, 9, 20, 21, 22, and 25. Numbers like 11, 12, 15, 18, and 19 are not counted because they are beyond n.

Thus, we've walked through this approach to find there are 9 good integers between 1 and 23.

Solution Implementation

Time and Space Complexity

The given code defines a function rotatedDigits which takes an integer n and returns the count of numbers less than or equal to n where the digits 2, 5, 6, or 9 appear at least once (making the number 'good'), and the digits 3, 4, or 7 do not appear at all (since they cannot be rotated to a valid number). It does not include numbers that remain the same after being rotated.

Time Complexity

The time complexity of the modified code is O(logN * 4^logN) which simplifies to O(N^2) where N is the number of digits in the input number n. This is because there are logN digits in n, and for each digit position, we iterate over up to 4 possible 'good' digits. There is a factor of logN for each digit position since there are that many recursive calls at each level, considering that limit is set to True only when i == up which means the next function call will honour the limit created by the previous digit.

Using memoization with @cache, we avoid repetitive calculations for each unique combination of the position of the digit, the flag whether we have encountered a 'good' digit, and whether we are limited by the original number's digit at this position (up) which leads to pruning the search space significantly.

Space Complexity

The space complexity of the code is O(logN * 2 * logN) which simplifies to O((logN)^2).

This includes space for:

• The memoization cache which could potentially store all states of the function arguments (pos, ok, limit).
• The array a of length 6, indicating the input number is decomposed into up to 6 digits, accounting for integers up to a maximum of 999999. However, this 6 is a constant and does not scale with the input, so it does not affect the time complexity.

Therefore, the space used by the recursion stack and the memoization dictionary depends on the number of different states the dfs function can be in, which is influenced by the number of digits logN (representing different positions) and the boolean flags ok and limit, leading to the complexity of O((logN)^2).

Learn more about how to find time and space complexity quickly using problem constraints.